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The problem asks for the exact value of \(\arcsin(-1)\).

The arcsine function, \(\arcsin(x)\), gives the angle whose sine is \(x\). The domain of \(\arcsin(x)\) is \([-1, 1]\), and its range is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). 

We are asked to find \(\arcsin(-1)\), which means we need to determine the angle \(\theta\) such that:

\[
\sin(\theta) = -1 \quad \text{and} \quad \theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right].
\]

The angle that satisfies this condition is \(\theta = -\frac{\pi}{2}\), because:

\[
\sin\left(-\frac{\pi}{2}\right) = -1.
\]

Thus, the exact value of \(\arcsin(-1)\) is:

\[
\arcsin(-1) = -\frac{\pi}{2}.
\]

The correct answer from the given options is:

\[
\boxed{-\frac{\pi}{2}}.
\]

Would you like further details or have any questions? Here are some related questions:

1. What is the value of \(\arcsin(1)\)?
2. How does the range of \(\arcsin(x)\) affect its outputs?
3. Can \(\arcsin(x)\) have values outside of \([- \frac{\pi}{2}, \frac{\pi}{2}]\)?
4. What is the relationship between \(\arcsin(x)\) and \(\sin(x)\)?
5. What is the value of \(\arcsin(0)\)?

**Tip:** The arcsine function is the inverse of the sine function, but only for values within its restricted domain, \([-1, 1]\).

Find the exact value of arcsin(-1) with a clear step-by-step solution. Learn about inverse trigonometric functions, the domain and range of arcsin(x), and why the answer is -π/2. Perfect for high school students studying trigonometry.

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